Passively mode locked quantum cascade lasers

ABSTRACT

This invention relates to a self-induced transparency mode-locked quantum cascade laser having an active section comprising a plurality of quantum well layers deposited in alternating layers on a plurality of quantum barrier layers and form a sequence of alternating gain and absorbing periods, said alternating gain and absorbing periods interleaved along the growth axis of the active section.

CROSS REFERENCE TO RELATED APPLICATIONS

The application claims priority benefit under 35 USC 119(e) to U.S.provisional patent application Ser. No. 61/144,178 filed 13 Jan. 2009,the contents of which are incorporated by reference herein in theirentirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Federal government funds were used in researching or developing thisinvention under National Science Foundation Grant No. EEC-0540832 underauspices of the MIRTHE Engineering Research Center.

NAMES OF PARTIES TO A JOINT RESEARCH AGREEMENT

None.

BACKGROUND

1. Field of the Invention

This invention relates to lasers that emit sub-picosecond pulses, andparticularly to semiconductor lasers called quantum cascade lasers thatemit in the mid- to far-infrared portion of the electromagneticspectrum. These pulses have particular importance to applications wherematerials must be processed while minimizing heat transfer and damage tothe materials, for example tissue removal in medicine and dentistry, andmicromachining at the pico and femto scales. These pulses also haveimportance to applications involving spectroscopy, such as detection oftrace compounds, clock synchronization in computer networks andtelecommunication networks, and in light detection and ranging (lidar).

2. Background of the Invention

Although interband lasers can be high-power lasers, it is difficult toobtain any power at all in interband lasers in the mid-IR. Quantumcascade lasers are semiconductor lasers that emit in the mid- tofar-infrared portion of the electromagnetic spectrum. Compared to othersemiconductor diode lasers that emit in the mid-far IR, QC lasers havehigher output power since laser emission is achieved from the transitionof an electron through periodic thin layers of material forming asuperlattice that introduces an electric potential over the device.Unlike other lasers, the lattice allows the electron to emit multiplephotons as it traverses, or cascades, from one period of the lattice tothe next. Semiconductor lattices in QC lasers can be made from layers ofcrystalline aluminum indium arsenide alternating with indium galliumarsenide, which create structures called quantum wells. One of theinteresting features of QC lasers is that they operate at roomtemperature, without the need for cooling that might be found in non-QClasers.

A technique for creating extremely short pulses of laser light is calledmode-locking. By pumping a laser into a laser cavity consisting of twoor more mirrors, the normal random oscillations of the light waves canbe made synchronous and thus increase (constructive interference),called mode-locking, or may be made to interfere and damp (destructiveinterference). By using an electronic device that modulates the lightintensity, an electrical signal can be used to establish a short pulsewithin the laser cavity in a technique called active mode-locking.However, in active mode-locking, pulses that are less than about onepicosecond have not been obtained. An actively modelocked QCL thatproduces 3 ps pulses at 6.2 μm was reported by Wang et al.

In contrast, passive mode-locking does not require an external signal tomaintain a continuous stream of pulses. Some examples of passivemode-locking use an intracavity absorber to absorb low-intensity lightand preferentially amplify high-intensity spikes. Certain dyes insolution can act as saturable absorbers, but graphite-latticestructures, and the use of an aperture have also been used to focus andamplify the high-intensity light and attenuate the low-intensity light.Aperture transmission of high-intensity light and attenuation oflow-intensity is called Kerr lens mode-locking, sometimes called “selfmode-locking” However, no semiconductor laser has been passivelymode-locked in the mid-IR range.

Laser light can have an interesting relationship to the medium ormaterial that it impacts. McCall and Hahn observed a nonlinear opticalpropagation effect in resonant medium. According to their observation, ashort pulse of coherent light above a critical input energy, for a givenpulse width τ<T₂, can pass through a saturable resonant medium as thoughthe medium were transparent. However, below the critical energy thisself-induced transparency (SIT) phenomenon cannot happen, rather, thepulse energy is absorbed.

Because QCLs typically have relatively short values of T₁ (1-10 ps) andrelatively long values of T₂ (˜100 fs) compared to other semiconductorlasers, standard passive modelocking cannot be achieved. However, theconditions are ideal for any approach, denoted SIT modelocking. Afterthe original observation of passive modelocking, it was speculated thatpassive modelocking was SIT modelocking, but this speculation was laterproved to be incorrect. Still later, Kozlov studied the possibility ofobtaining SIT modelocking by using gas lasers with two separate tubes inwhich the gases in the two different tubes have the same resonance linesand the gas in one tube has twice the dipole moment of the other.However, this design cannot be implemented in practice. Applications formode-locked QC lasers include three-dimensional micro-, ornano-fabrication or machining, investigative and diagnosticspectroscopy, optical communication and electronic devices such aslimiters or storage, and medical and dental probes and surgical devices.

BRIEF SUMMARY OF THE INVENTION

To address limitations in the prior art, there is provided in onepreferred embodiment a self-induced transparency mode-locked quantumcascade laser, comprising:

(i) an active section comprising a plurality of quantum well layersdeposited in alternating layers on a plurality of quantum barrier layersand form a sequence of alternating gain and absorbing periods, saidalternating gain and absorbing periods interleaved along the growth axisof the active section, wherein the absorbing periods have a dipolemoment of about twice that of the gain periods;(ii) an optical cavity that houses the active section and permitsamplified light to escape;(iii) an externally supplied seed pulse; and(iv) current injectors structured and arranged to apply an electriccontrol field to the active section.

In another preferred embodiment, there is also provided wherein thequantum barrier layers are made using Indium-Aluminum-Arsenide, and thequantum well layers are made using Indium-Gallium-Arsenide.

In another preferred embodiment, there is also provided wherein the gainand absorbing periods are designed to provide a mid-IR wavelength laser.

In another preferred embodiment, there is also provided wherein the gainand absorbing periods are designed to provide a mid-IR wavelength laserhaving a wavelength of between about 3 micrometers and about 12micrometers, and in some embodiments between about 8 and about 12micrometers.

In another preferred embodiment, there is also provided wherein the gainand absorbing periods are designed to provide coherence times

gain recovery times

round trip times.

In another preferred embodiment, there is also provided wherein the gainand absorbing periods are designed to provide: coherence times of atleast 50 to about 200 femtoseconds, and in one embodiment about 100femtoseconds; gain recovery times of at least 1 to about 10 picoseconds,and in one embodiment about 1 picosecond; and round trip times of atleast 40-60 picoseconds, in one embodiment 50 picoseconds.

In another preferred embodiment, there is also provided wherein the gainand absorbing periods are designed to provide a heterostructure having aratio of 4 gain periods:1 absorbing period. In other embodiments, theheterostructure may include a ratio of 3 gain periods to 1 absorbingperiod, or 5 gain periods to 1 absorbing period.

In another preferred embodiment, there is also provided wherein theinput pulse is a it pulse in the gain medium.

In another preferred embodiment, there is also provided wherein eachgain period and each absorbing period have over 16 quantum layers.

In another preferred embodiment, there is also provided wherein the gainand absorbing periods are designed to suppress continuous waves andeliminate spatial hole burning.

In another preferred embodiment, there is also provided wherein the gainand absorbing periods are designed to provide a mid-IR wavelength laserhaving pulse length less than 100 femtoseconds long.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a Schematic of a QCL structure with gain and absorbingperiods. On the left, we show a cutaway view of the QCL structure. Theactive region is shown as a filled-in rectangle. We are looking in thedirection along which light would propagate. Electrodes would be affixedto the top and bottom so that electrons flow vertically. Theheterostructure would also be stacked vertically as shown on the right.We show one absorbing period for every four gain periods, correspondingschematically to the case in which the electron density in the gainmedium N_(g)≅4× the electron density in the absorbing medium (N_(a)),and we show absorbing periods that are twice as large as gain periods toindicate schematically that the dipole moment in the absorbing medium(μ_(a))≅2× the dipole moment in the gain medium (μ_(g)).

FIG. 2. Schematic of the (a) gain and (b) absorbing media. Blackstraight-line arrows indicate the direction of electron flow. Red wavyarrows indicate radiative transitions.

FIG. 3. Conduction-band diagram and moduli-squared wave functions forone gain and one absorbing period of the (a) 12 μm, (b) 8 μmmode-locking QCL structures. The sequence of layer dimensions is (in Å,starting from left): (a) 37, 36, 10, 35, 10, 34, 11, 34, 12, 35, 39, 37,12, 62, 14, 58, 28, 42, 12, 40, 13, 37, 15, 34, 15, 34, 34, 45, 11, 65,6, 69; (b) 42, 34, 9, 33, 12, 30, 13, 28, 16, 28, 41, 27, 18, 62, 14,58, 28, 42, 12, 40, 13, 37, 13, 34, 16, 34, 34, 9, 31, 50, 5, 84. Thenumbers in bold type indicate In_(0.52)Al_(0.48)As barrier layers andthose in roman type indicate In_(0.53)Ga_(0.47)As well layers. Red wavyarrows indicate radiative transitions.

FIG. 4. Stability limits of the normalized absorption (ā) vs thenormalized gain ( g) coefficients with different Δ_(g0) and Δ_(a0). Theratio T₁/T₂ is infinity in all cases. For a given τ and ā, the requiredg increases as Δ_(g0) and |Δ_(a0)| decrease.

FIG. 5. Stability limits of the normalized absorption (ā) vs thenormalized gain ( g) coefficients with different T_(2a)/T_(2g). We setT_(1g)=T_(1a)=∞ in all cases. In equilibrium, the gain medium iscompletely inverted, i.e., Δ_(g0)=1.0, and the absorbing medium iscompletely uninverted, i.e., Δ_(a0)=−1.0.

FIG. 6

Stability limits of the normalized absorption vs the normalized gaincoefficients with different T₁/T₂. In equilibrium, the gain medium iscompletely inverted, i.e., Δ_(g0)=1.0 and the absorbing medium iscompletely uninverted, i.e., Δ_(a0)=−1.0. In each bundle of dashedlines, corresponding to a fixed value of τ, T₁/T₂ decreases from left toright.

FIG. 7

Pulse evolution in the system. (a) Gain and absorption coefficients arein the stable regime, g=4.0, ā=3.5. (b) Gain and absorption coefficientsare in the regime in which continuous waves are unstable, g=4.0, ā=1.0.(c) Gain and absorption coefficients are in the regime where any initialpulse attenuates, g=4.0, ā=7.8. The ratio T₁/T₂ equals 10 in all cases.In equilibrium, the gain medium is completely inverted, i.e.,Δ_(g0)=1.0, and the absorbing medium is completely uninverted, i.e.,Δ_(a0)=−1.0.

FIG. 8

Stability limits of the normalized absorption (ā) vs the normalized gain( g) coefficients with different values of T_(1a)/T_(1g). We setT_(2g)=T_(2a) and T_(1g)/T_(2g)=10 in all cases. In equilibrium, thegain medium is completely inverted, i.e., Δ_(g0)=1.0, and the absorbingmedium is completely uninverted, i.e., Δ_(a0)=−1.0

FIG. 9

Stability limits of the ratio of the dipole moments in the absorbing andgain media ( μ) vs the normalized gain coefficient ( g) for three casesof normalized absorption (ā). The ratio T₁/T₂ is 10 in all cases. Inequilibrium, the gain medium is completely inverted, i.e., Δ_(g0)=1.0,and the absorbing medium is completely uninverted, i.e., Δ_(a0)=−1.0

FIG. 10

Input pulse energy limits vs normalized input pulse duration (τ_(i)/T₂)for two different cases of g and ā. In both the cases, we set T₁/T₂=10.In equilibrium, the gain medium is completely inverted, i.e.,Δ_(g0)=1.0, and the absorbing medium is completely uninverted, i.e.,Δ_(a0)=−1.0.

DETAILED DESCRIPTION OF THE INVENTION

Quantum cascade lasers (QCLs) are important light sources in themidinfrared range, about 3 μm to about 12 μm. The light is generated bya transition between two subbands in the conduction band, in contrast tointerband semiconductor lasers. As a consequence, the subbands havenarrow linewidths and long coherence times T₂ compared to interbandsemiconductor lasers. Values of T₂ on the order of 100 fs areachievable. Another important feature of the QCLs is their rapid gainrecovery times T₁ compared to interband semiconductor lasers due to fastLO-phonon relaxation. Typical values of T₁ are in the range 1-10 ps,which is short compared to T_(rt), the round-trip time in the cavity.Typical values of T_(rt) are on the order of 50 ps. The narrowlinewidths and fast recovery times of QCLs make it difficult to achieveconventional passive mode locking. Gain bandwidths that aresignificantly larger than the pulse bandwidths are required, and that ishard to obtain when the linewidths are narrow, as in QCLs. A saturablegain, with a recovery time that is long compared to T_(rt), is alsorequired for conventional passive mode locking in order to suppresscontinuous waves, and the typical gain recovery times in QCLs areshorter than the round-trip times. Thus, conventional passive modelocking cannot work in QCLs that operate in a standard parameter regime.However, QCLs are an ideal tool for creating the long-predicted,never-observed SIT mode locking. The relationship T₂

T₁

T_(rt), which is typical for QCLs, is precisely what is needed for SITmode locking, and the ease of band-gap engineering in QCLs makes itpossible to interleave gain and absorbing periods as needed. Conversely,SIT mode locking of QCLs makes it possible to obtain mode-locked pulsesfrom a mid-infrared laser that are less than 100 fs in duration. UsingSIT mode-locking in a QC laser also addresses the stability problemsrelating to the growth of continuous waves and spatial hole burning. InSIT, continuous waves are suppressed and mode-locked pulses propagate inone direction or the other at a time so that spatial hole burning is notan issue.

Referring now to the FIGURES, specific non-limiting examples andpreferred embodiments of SIT mode-locking structures for QC lasers aredescribed.

In order to obtain SIT mode locking, it is necessary to have two highlycoherent resonant media with nearly equal resonant frequencies. In onemedium, denoted the gain medium, electrons should be injected into theupper lasing state so that the resonant states are nearly inverted. Inthe other medium, denoted the absorbing medium, electrons should beinjected into the lower state so that the resonant states are notinverted. Also, the dipole strength in the absorbing medium should benearly equal to twice the dipole strength in the gain medium. At thesame time, the ratio of the gain per unit length to the absorption perunit length should be small enough so that the growth of continuouswaves is suppressed, but large enough so that a mode-locked pulse canstably exist. It is possible to simultaneously satisfy all theseconditions by interleaving gain and absorbing periods that have therequired dipole strengths as shown schematically in FIG. 1.

FIG. 1 shows a schematic of a QCL structure with gain and absorbingperiods. On the left, we show a cutaway view of the QCL structure. Theactive region is shown as a filled-in rectangle. We are looking in thedirection along which light would propagate. Electrodes would be affixedto the top and bottom so that electrons flow vertically. Theheterostructure would also be stacked vertically as shown on the right.We show one absorbing period for every four gain periods, correspondingschematically to the case in which the electron density in the gainmedium (N_(g))≅4× the electron density in the absorbing medium (N_(a)),and we show absorbing periods that are twice as large as gain periods toindicate schematically that the dipole moment in the absorbing medium(μ_(a))≅2× the dipole moment in the gain medium (μ_(g)). Byappropriately choosing the number of gain periods and the number ofabsorbing periods, one can obtain any desired ratio for the gain andabsorption per unit length. As long as there are many periods within thetransverse wavelength of the lasing mode, the gain and absorbing periodswill experience the same light intensity.

In FIG. 2, we show simplified two-level resonant structures for the gainand absorbing media. FIG. 2 shows a schematic of the (a) gain and (b)absorbing media. Black straight-line arrows indicate the direction ofelectron flow. Red wavy arrows indicate radiative transitions. In thegain medium, electrons are injected into level 2 g and are extractedfrom level 1 g. The carrier lifetime in 2 g should be longer than themode-locked pulse duration and the equilibrium population inversionshould be nearly complete. When an optical pulse with a photon energyequal to the resonant energy impinges on the gain medium with itspolarization oriented in the direction perpendicular to the layers,electrons scatter to level 1 g and photons are emitted. Then, theelectrons are nonradiatively extracted from level 1 g. In the absorbingmedium, electrons are injected into the lower level 1 a. The lifetime ofstate 1 a should again be longer than the pulse width. When a lightpulse of the appropriate wavelength and polarization impinges on theabsorbing medium, photons are absorbed and electrons jump to level 2 a.If a light pulse has enough intensity, then photons are re-emitted withno overall loss in one Rabi oscillation time. In order for theseprocesses in the gain medium and absorbing medium to occursimultaneously, the energy spacing between the resonant levels should benearly the same in both media.

In the theory of resonant two-level media, both π pulses and 2π pulsesplay an important role. A π pulse is a pulse with sufficient energy toexactly invert the lower state population of a two-level medium if themedium is initially uninverted, or, conversely, to uninvert the upperstate if the medium is initially inverted. In the former case, the pulseexperiences loss and rapidly attenuates, but, in the latter case, thepulse experiences gain. The pulse duration is approximately half a Rabioscillation period. If a pulse lasts a longer time than required todrive the population from the upper level to the lower, then the mediumwill amplify the first part of the pulse and attenuate the latter part,in a way that shortens the pulse. Conversely, if a pulse is initiallytoo short, it is lengthened. Because a π pulse experiences gain, it isnatural that shortly after the initial observations of passive modelocking in lasers, it was proposed that the pulses in these lasers areactually SIT-induced π pulses. However, these pulses are not suitablefor use on their own as passively mode-locked laser pulses. Where one πpulse can exist, there is nothing to prevent continuous waves fromgenerating multiple pulses leading to chaos rather than a single stablepulse oscillating in a laser cavity.

One can circumvent this difficulty by combining a gain medium in whichthe optical pulse is a π pulse with an absorbing medium in which theoptical pulse is a 2π pulse. A 2π pulse is a pulse with sufficientenergy so that in an uninverted medium the lower state population isfirst inverted and then returned to the lower state in approximately oneRabi oscillation. A 2π pulse like a π pulse is stable. If its initialduration is too long, the duration decreases, and, if its initialduration is too short, the duration increases. The 2π pulse propagatesthrough the medium without loss, in contrast to continuous waves at theresonant optical frequency, which experience loss. This remarkableproperty is what led to the name “self-induced transparency.” Because ofthis property, the absorbing medium acts like the saturable loss in aconventional passively modelocked system, suppressing the growth ofcontinuous waves, while allowing a short pulse to propagate.

It is evidently important that both the gain medium and the absorbingmedium act on the optical pulse simultaneously. We may achieve thissimultaneous interaction by designing a QCL structure that has the gainand absorbing periods interleaved along the growth axis of thestructure, as shown in FIG. 1. By making the dipole moment in theabsorbing periods twice that of the gain periods, a π pulse in the gainperiods is a 2π pulse in the absorbing periods. Therefore, an injected πpulse completely depletes the gain medium as it makes its way throughthe QCL, whereas, the absorbing medium becomes transparent. We will showthat by controlling the amount of gain and absorption per unit length inthe gain and absorbing media, pulse durations can be controlled.

In order to suppress spatial hole burning, the RNGH instability, or thegrowth of multiple pulses, we do not want continuous waves to grow in anSIT mode-locked laser. The absorption parameter should be large enoughto keep the laser operating below the threshold for the growth ofcontinuous waves. Therefore, the laser cannot self-start and it isnecessary to use some means to start the mode locking. Essentially, weneed to create an optical pulse that has sufficient energy and aduration on the order of T₁. This pulse can be created by using anelectrical shock, electrical pulse, a mechanical vibration or anexternal optical seed. We suggest two optical approaches. First, we canseed the pulse from an external source by injection locking, or, second,we can use active mode locking to generate an initial pulse that willhave a suitable energy and initial duration for SIT mode locking. It mayalso be possible to use a mechanical or an electrical impulse to startthe mode locking.

Quantum Cascade Laser Structures

We have designed QCL gain and absorbing periods that fulfill the SITmode locking requirements at two different wavelengths, 8 and 12 μm.Similar structures can be designed over a broad range of mid-IRwavelengths, from about 3 μm to about 12 μm. We use theIn_(0.52)Al_(0.48)As/In_(0.53)Ga_(0.47)As material system for the activeregion in our design since most of the demonstrated QCLs to date havebeen based on this material system. However, mode-locking structuresoperating at less than 8 μm will be difficult to design using thismaterial system. Since electrons are injected into the lower state inthe absorbing medium, the upper state approaches close enough to theconduction-band edge at wavelengths below 8 μm to lead to a largeincrease in the carrier loss due to scattering to the continuum-likestates. To design a mode-locking structure at shorter wavelengths, itshould be possible to use a strain-balanced InAlAs/InGaAs materialsystem.

The gain periods in our design are typical QCL periods. We design athree-quantum-well active region for the gain periods that has adiagonal transition, which lowers the dipole moment relative to designsthat have a vertical transition. This choice simplifies the design ofthe absorbing periods. Population inversion is achieved by confining theresonant states in separate quantum wells and depopulating the lowerstate through phonon relaxation to another state sitting below the lowerresonant state. While the dipole moment in the gain periods is not high,the upper state lifetime is larger than in the case of verticaltransitions so that the gain remains high. The design of absorbingperiods is different from the design of gain periods. The combinedrequirements of carrier injection into and extraction from the lowerresonant level and a dipole moment twice that of the gain periods makeit difficult to design the absorbing periods. To achieve a large dipolemoment, the transition should be between two excited states. The carrierlifetime is made high by reducing scattering through phonon relaxationand reducing the carrier tunneling from the lower resonant state intothe succeeding injector states. The injector regions have differentdesigns when the electrons are tunneling into a gain or absorbing activeregion due to the different quantum electronic structures of gain andabsorbing active regions.

The structure in FIG. 3( a) emits light at 12 μm. FIG. 3.Conduction-band diagram and moduli-squared wave functions for one gainand one absorbing period of the (a) 12 μm, (b) 8 μm mode-locking QCLstructures. The sequence of layer dimensions is (in Å, starting fromleft): (a) 37, 36, 10, 35, 10, 34, 11, 34, 12, 35, 39, 37, 12, 62, 14,58, 28, 42, 12, 40, 13, 37, 15, 34, 15, 34, 34, 45, 11, 65, 6, 69; (b)42, 34, 9, 33, 12, 30, 13, 28, 16, 28, 41, 27, 18, 62, 14, 58, 28, 42,12, 40, 13, 37, 13, 34, 16, 34, 34, 9, 31, 50, 5, 84. The numbers inbold type indicate In_(0.52)Al_(0.48)As barrier layers and those inroman type indicate In_(0.53)Ga_(0.47)As well layers. Red wavy arrowsindicate radiative transitions.

Electrons are injected into level 3 g, and the gain transition isbetween levels 3 g and 2 g. The dipole moment between the resonantlevels is given by μ_(g)/e=1.81 nm. Level 3 g has a lifetime of ˜2 ps.Level 1 g is positioned approximately at phonon resonance with 2 g.Level 2 g has a lifetime of ˜0.5 ps, so that the population inversion ishigh. The transition energy is 101 meV. In the absorbing periods,electrons are injected into level 4 a and they jump to level 5 a byabsorbing photons. The lifetime of level 5 a is ˜0.83 ps. The absorbinglevels are separated by 101 meV and have a dipole moment μ_(a)/e=3.65nm. After electrons are injected into level 3 g, they have a longlifetime of 2 ps due to the low scattering rate. When the optical pulsearrives, the population in level 3 g decreases to level 2 g. Level 2 gis depopulated quickly to level 1 g through phonon interactions and theelectrons then transit to the following injector stage. Populationinversion is restored before the optical pulse makes a round trip in thelaser cavity. Electrons sit in level 4 a after being injected by thepreceding injector stage. When an optical pulse reaches the absorbingperiods, electrons from level 4 a move to level 5 a by absorbingphotons. Since the pulse is a 2π pulse for the absorbing medium, level 5a is depleted by making a Rabi oscillation, during which photons areemitted.

In the structure given in FIG. 3( b), the gain transition is betweenlevels 3 g and 2 g. The dipole moment between the gain levels is 1.55 nmand the lifetime of level 3 g is −3 ps. The resonant transition energyis 150 meV in both the gain and absorbing periods. In this structure,electrons are injected into level 3 a in the absorbing periods and theabsorbing transition is between levels 3 a and 4 a. Level 4 a has alifetime of ˜0.8 ps. The absorbing levels have a dipole momentμ_(a)/e=2.95 nm.

In a QCL, the time T₁ is determined mainly by the LO-phonon relaxationrate. The LO-phonon relaxation rate depends mainly on the energy spacingbetween the levels and the overlap of the corresponding wave functions,so that T₁ depends on the details of the band structure and can varygreatly from design to design. Indeed, strictly speaking there is not asingle T₁ in either the gain or the absorbing medium since there aremore than two levels involved in the dynamics in both media. In the gainstage of the QCL structure given in FIG. 3, the upper state is confinedin a separate quantum well from the other states in the active region,so that the phonon relaxation rate is smaller than when all the activestates are in the same well and the lifetime is higher. However, theabsorbing stage is designed such that the upper and lower state wavefunctions have a large overlap, which makes the dipole moment higher,but decreases the lifetime. Therefore, in practical designs, we findT_(1g)>T_(1a).

Generally, if the gain and absorbing media are grown from the samematerial system, it is reasonable to assume T_(2g)≈T_(2a). In the QCLstructure that we propose, an In_(0.52)Al_(0.48)As/In_(0.53)Ga_(0.47)Asmaterial system is used for both the gain and absorbing periods, so thatT_(2a) should not vary much from T_(2g). However, there has yet to be adetailed theoretical calculation of these coherence times. In a QCL, thevalue of T₂ depends mainly on electron-electron scattering, electron-LOphonon scattering, and interface scattering. Therefore, the values ofT_(2g) and T_(2a) may differ somewhat, depending on the details of thedesign.

Maxwell-Bloch Equations

Wang et al. and Gordon et al. have observed the RNGH instability in QCLswith only gain periods. They showed evidence for Rabi oscillations anddemonstrated that the two-level Maxwell-Bloch equations apply to QCLs insome parameter regimes, although they also showed that saturableabsorption affects the behavior quantitatively, significantly reducingthe RNGH threshold. Gordon et al. attributed the saturable absorption toKerr lensing that increases the mode overlap with the active region andreduces the wall losses. These effects depend sensitively on the detailsof the QCL geometry. They also observed that spatial hole burning due tothe interaction of forward- and backward-propagating waves has animportant effect on the pulse spectrum. They did not find it necessaryto include chromatic dispersion or other nonlinearities. Motivated bythese results, we use a simple two-level model based on the standardone-dimensional Maxwell-Bloch equations. The Maxwell-Bloch equationsthat describe the light propagation and light-matter interaction in QCLhaving interleaved gain and absorbing periods may be written as

$\begin{matrix}{{{\frac{n}{c}\frac{\partial E}{\partial t}} = {{- \frac{\partial E}{\partial z}} - {\frac{{kN}_{g}\Gamma_{g}\mu_{g}}{2ɛ_{0}n^{2}}\eta_{g}} - {\frac{{kN}_{a}\Gamma_{a}\mu_{a}}{2ɛ_{0}n^{2}}\eta_{a}} - {\frac{1}{2}{lE}}}},} & \left( {1a} \right) \\{\frac{\partial\eta_{g}}{\partial t} = {{\frac{\; \mu_{g}}{2\hslash}\Delta_{g}E} - \frac{\eta_{g}}{T_{2g}}}} & \left( {1b} \right) \\{\frac{\partial\Delta_{g}}{\partial t} = {{\frac{\; \mu_{g}}{\hslash}\eta_{g}E^{*}} - {\frac{\; \mu_{g}}{\hslash}\eta_{g}^{*}E} + \frac{\Delta_{g\; 0} - \Delta_{g}}{T_{1g}}}} & \left( {1c} \right) \\{\frac{\partial\eta_{a}}{\partial t} = {{\frac{\; \mu_{a}}{2\hslash}\Delta_{a}E} - \frac{\eta_{a}}{T_{2a}}}} & \left( {1d} \right) \\{\frac{\partial\Delta_{a}}{\partial t} = {{\frac{\; \mu_{a}}{\hslash}\eta_{a}E^{*}} - {\frac{\; \mu_{a}}{\hslash}\eta_{a}^{*}E} + \frac{\Delta_{a\; 0} - \Delta_{a}}{T_{1a}}}} & \left( {1e} \right)\end{matrix}$

where the subscripts g and a in Eq. (1) represent gain and absorption,respectively. The independent variables z and t are length along thelight-propagation axis of the QCL and time. The dependent variablesE,η_(g,a), and Δ_(g,a) refer to the envelopes of the electric field,gain polarization, and gain inversion. The parameters Δ_(g0) and Δ_(a0)refer to the equilibrium inversion away from the mode-locked pulse. Theparameters n and c denote the index of refraction and the speed oflight. The parameters N_(g,n),Γ_(g,a), denote the effective electrondensity multiplied by the overlap factor. The parameters k, l, ∈₀, and hdenote the wave number in the active region, the linear loss includingthe mirror loss, the vacuum dielectric permittivity, and Planck'sconstant. The notation closely follows that of Wang et al., with thedifferences that we have an absorbing as well as a gain medium, and weare considering unidirectional propagation, as is appropriate for amode-locked pulse.

In order to achieve SIT mode locking, the growth of continuous wavesmust be suppressed. At this point, two observations are made. First,because continuous waves are suppressed, forward- andbackward-propagating waves cannot interact, and spatial hole burningwill not occur. Second, we did not include saturable absorption in Eq.(1) because we are not certain how to do so. This contribution was addedphenomenologically to the Maxwell-Bloch equations by Wang et al. andGordon et al., based on experimental observations in particular QCLs andwas attributed to effects that depend sensitively on the geometry ofthose QCLs.

In order to suppress continuous waves, the gain must be below threshold.To derive this condition, we set Δ_(g)=Δ_(g0) and Δ_(a)=Δ_(a0) in steadystate, where there is no evolution in z. We then find from Eqs. (1b) and(1d),

$\begin{matrix}{{\eta_{g} = {\frac{\mu_{g}}{2\hslash}T_{2g}\Delta_{g\; 0}E}},{\eta_{a} = {\frac{\mu_{a}}{2\hslash}T_{2a}\Delta_{a\; 0}E}},} & (2)\end{matrix}$

where we are considering continuous waves, so that there is nodependence on t and the t-derivatives vanish. After substitution in Eq.1a, we obtain in steady state

$\begin{matrix}{{{\frac{{kN}_{g}\Gamma_{g}\mu_{g}^{2}T_{2g}\Delta_{g\; 0}}{2ɛ_{0}n^{2}\hslash} + \frac{{kN}_{a}\Gamma_{a}\mu_{a}^{2}T_{2a}\Delta_{a\; 0}}{2ɛ_{0}n^{2}\hslash} - l} = 0},} & (3)\end{matrix}$

which may also be written as gΔ_(g0)+aΔ_(a0)−l=0, where

$\begin{matrix}{{g = \frac{{kN}_{g}\Gamma_{g}\mu_{g}^{2}T_{2g}}{2ɛ_{0}n^{2}\hslash}},\mspace{14mu} {a = {\frac{{kN}_{a}\Gamma_{a}\mu_{a}^{2}T_{2a}}{2ɛ_{0}n^{2}\hslash}.}}} & (4)\end{matrix}$

Physically, the parameter g corresponds to the gain per unit length fromthe gain periods of the QCL and the parameter a corresponds to theabsorption per unit length from the absorbing periods. The condition forthe linear gain to remain below threshold is gΔ_(g0)+aΔ_(a0)−l<0. In thecase of a fully inverted gain medium, so that Δ_(g)=Δ_(g0)=1 and a fullyuninverted absorbing medium so that Δ_(a)=Δ_(a0)=−1, the condition tosuppress continuous waves becomes g−a−1<0.

Assuming that T_(1g) and T_(1a) are large enough so that they may be setequal to co in Eq. 1, and focusing on the special case in whichμ_(a)=2μ_(g) and the pulse is a it pulse in the gain medium, Eq. 1 hasan exact analytical solution that we may write as

$\begin{matrix}{{E = {\frac{\hslash}{\mu_{g}\tau}{sech}\; x}},} & \left( {5a} \right) \\{{\eta_{g} = {\frac{\; B_{g}}{2}\Delta_{g\; 0}{sech}\; x}},} & \left( {5b} \right) \\{{\Delta_{g} = {B_{g}{\Delta_{g\; 0}\left( {\frac{\tau}{T_{2\; g}} - {\tanh \; x}} \right)}}},} & \left( {5c} \right) \\{{\eta_{a} = {\frac{\; B_{a}}{2}{\Delta_{a\; 0}\left( {{{- {sech}}\; x\; \tanh \; x} + {\frac{\tau}{3T_{2a}}{sech}\; x}} \right)}}},} & \left( {5d} \right) \\{{\Delta_{a} = {{\frac{B_{a}}{2}{\Delta_{a\; 0}\left( {1 + \frac{\tau^{2}}{3T_{2a}^{2}}} \right)}} - {B_{a}{\Delta_{a\; 0}\left( {{{sech}^{2}x} + {\frac{2\tau}{3T_{2a}}\tanh \; x}} \right)}}}},{where}} & \left( {5\; e} \right) \\{{x = {\frac{t}{\tau} - \frac{z}{v\; \tau}}},{while}} & (6) \\{{B_{g} = \frac{1}{1 + {\tau/T_{2g}}}},\mspace{14mu} {{{and}\mspace{14mu} B_{a}} = \frac{2}{\left( {1 + {{\tau/3}T_{2a}}} \right)\left( {1 + {\tau/T_{2a}}} \right)}}} & (7)\end{matrix}$

are chosen such that Δ_(g)→Δ_(g0) and Δ_(a)→Δ_(a0) as t→−∞, so that theequilibrium population completely recovers on every pass of the pulsethrough the laser. Hence, Eq. 5 shows that stable mode-locked operationcan be achieved in the proposed structure. The parameters τ and ν thatcorrespond to the pulse duration and the pulse velocity are determinedby the equations

$\begin{matrix}{{{{\frac{\tau/T_{2g}}{1 + {\tau/T_{2g}}}g\; \Delta_{g\; 0}} + {\frac{{\tau^{2}/3}T_{2a}^{2}}{\left( {1 + {\tau/T_{2a}}} \right)\left( {1 + {{\tau/3}T_{2a}}} \right)}a\; \Delta_{a\; 0}} - l} = 0}{and}} & (8) \\{\frac{1}{v} = {\frac{n}{c} - {a\; \tau \frac{{\tau/2}T_{2g}}{\left( {1 + {\tau/T_{2a}}} \right)\left( {1 + {{\tau/3}T_{2a}}} \right)}{\Delta_{a\; 0}.}}}} & (9)\end{matrix}$

The full-width half-maximum pulse duration τ_(FWHM) equals 1.763τ. Thissolution was previously presented in the special case Δ_(g0)=1.0 andΔ_(a0)=−1.0. We now consider in more detail the special caseT_(2g)=T_(2a)≡T₂. Writing g=g/l, ā=a/l, and τ=τ/T₂, we find that thecondition to suppress the growth of continuous waves becomesgΔ_(g0)+āΔ_(a0)−1<0, and the equation for the pulse duration becomes 10,and 11

$\begin{matrix}{\frac{3}{\overset{\_}{\tau}} = {\frac{{3\overset{\_}{g}\; \Delta_{g\; 0}} - 4}{2} + {\left\lbrack {\left( \frac{{3\overset{\_}{g}\; \Delta_{g\; 0}} - 4}{2} \right)^{2} + {3\left( {{\overset{\_}{g}\; \Delta_{g\; 0}} + {\overset{\_}{a}\; \Delta_{a\; 0}} - 1} \right)}} \right\rbrack^{1/2}.}}} & (10)\end{matrix}$

Equation (10) only has a solution when ā<(3 gΔ_(g0)−2)²/12|Δ_(a0)|,where we note that Δ_(a0)<0. The conditions for stability may besummarized as

$\begin{matrix}{\frac{\left( {{\overset{\_}{g}\; \Delta_{g\; 0}} - 1} \right)}{\Delta_{a\; 0}} < \overset{\_}{a} < {\frac{\left( {{3\overset{\_}{g}\; \Delta_{g\; 0}} - 2} \right)^{2}}{12{\Delta_{a\; 0}}}.}} & (11)\end{matrix}$

When ā is above the upper limit in Eq. (11), we have found by solvingEq. (1) computationally that an initial pulse damps away. When ā isbelow the lower limit, continuous waves grow. We have computationallyfound that multiple pulses are generated in this case.

Equation (11) defines a parameter regime in which stable mode-lockedoperation is possible. In FIG. 4, we present the stability limits whenthe population inversion in the gain and absorbing periods vary. In allcases, the lower lines indicate the limiting values for ā, below whichcontinuous waves grow, and the upper lines indicate the limiting valuesfor ā, above which initial pulses damp. FIG. 4 shows that the minimumvalue of g that is required for stable operation increases when Δ_(g0)decreases and Δ_(a0) increases by the same amount. There is also aslight decrease in the lower limit for ā and a larger decrease in theupper limit. Since the upper limit drops more than the lower limit, thestable parameter region becomes smaller. We also show contours of thepulse duration, normalized by the coherence time T_(2g), denoted τ, asgiven by Eq. (10). Pulse durations are approximately on the order ofT_(2g) when g≈2.5 and ā≈2.0. We also note that pulses become shorter asg and ā increase. However, both g and ā are directly proportional to thecurrent; so, to increase the gain and absorption in a fixed ratio, onemust increase the current. At the same time, we note that g and ā aredirectly proportional to T₂. Hence, it is possible to reduce therequired current by increasing T₂.

We have studied what happens to the stability limits if T_(2a)/T_(2g)vary and we show the results in FIG. 5. In FIG. 5, we have varied T_(2a)keeping T_(2g) constant. In a QCL, typical values of T_(2g) and T_(2a)are on the order of 100 fs. A change in T_(2a) affects the stabilitylimits more than does a change in T_(2g) as is evident from Eq. 8. WhenT_(2a)/T_(2g) increases, the upper stability limits increase. Wheng=4.0, we find that the upper limit for ā varies from 3.75 to 8.3 to 24as T_(2a)/T_(2g) varies from 0.5 to 1.0 to 2.0. The lower limit for āremains unchanged.

We now derive an energy-balance equation that describes the energy inputlimits for stable operation when τ

T₂. We define Θ(z,t)=∫_(−∞) ^(t)E(z,t′)dt′. Then, Eqs. (1b) and (1c) canbe written as

$\begin{matrix}{{\frac{\partial{\eta_{g}\left( {z,t} \right)}}{\partial t} = {\frac{\; \mu_{g}}{2\hslash}{\Delta_{g}\left( {z,t} \right)}\frac{\partial{\Theta \left( {z,t} \right)}}{\partial t}}}{and}} & (12) \\{\frac{\partial{\Delta_{g}\left( {z,t} \right)}}{\partial t} = {2\frac{\; \mu_{g}}{\hslash}{\eta_{g}\left( {z,t} \right)}{\frac{\partial{\Theta \left( {z,t} \right)}}{\partial t}.}}} & (13)\end{matrix}$

In the gain medium, the polarization and population inversion can bewritten in terms of a single angle α as 2iη_(g)=cos α and Δ_(g)=sin α.We integrate both sides of Eq. (12) or Eq. (13), after substitutingthese expressions for η_(g) and Δ_(g) and assuming that Δg(z,t→−∞)=1. Wethen obtain α(z,t)=π/2+(μ_(g)/ h)Θ(z,t). We may similarly write2iη_(a)=cos β and Δ_(a)=sin β in the absorbing medium, and we then findβ(z,t)=−π/2+(μ_(a)/ h)Θ(z,t), where we have set Δ_(a)(z,t→−∞)=−1. We nowconsider Eq. (1a) in steady state, where there is no evolution in z, andin the limit t→∞, where there is no evolution in t. We also define anormalized pulse energy

θ(z)=(μ_(g) / h )Θ(z,t→∞).  (14)

Equation (1a) now becomes

$\begin{matrix}{{{g\; {\sin \left\lbrack {\overset{\_}{\theta}(z)} \right\rbrack}} = {a\; \frac{\mu_{g}}{\mu_{a}}\frac{T_{2g}}{T_{2a}}{\sin \left\lbrack {\frac{\mu_{a}}{\mu_{g}}{\overset{\_}{\theta}(z)}} \right\rbrack}}},} & (15)\end{matrix}$

where we note that the linear loss may be neglected in the limit τ<<T₂.In the special case μ_(a)=2μ_(g), and T_(2g)=T_(2a)≡T₂, we finda(μ_(g)/μ_(a))(T_(2g)/T_(2a))sin [(μ_(a)/μ_(g)) θ]=a cos θ sin θ, sothat Eq. (15) becomes cos θ=g/a, which defines the limits of the inputenergy that is required to generate a single pulse,

cos⁻¹(g/a)< θ<2π−cos⁻¹(g/a).  (16)

When the initial value of θ is within these limits, a single pulse witha final value of θ=π is generated. When the initial value of θ is belowthis value, the lower limit in Eq. (16), the initial pulse damps. Whenthe initial value of θ is above the upper limit, the initial pulsesplits into multiple pulses.

In the analysis up to now it has been assumed that the central carrierfrequency of the light pulse and the transition frequency in both thegain and absorbing media are the same. Since the frequency of the lightis largely determined by the gain medium, it is reasonable to assumethat there is no detuning between the light and the gain medium. Even ifthe mode locking is seeded by injection locking, analogous to theexperiment of Choi et al. the injection-locking laser can be tuned tothe gain resonance. There may be a small detuning between the gain andabsorbing media due to design or growth issues; however, it is possibleto design the gain and absorbing media so that detuning is nearlyabsent. QCLs are currently being grown with high accuracy andexperimentally observed wavelengths agree closely with the designedvalues. If there is a detuning of Δω between the gain and the absorbingperiods and the light pulses are tuned to the gain periods, Eq. 1dbecomes

$\begin{matrix}{\frac{\partial\eta_{a}}{\partial t} = {{\frac{\; \mu_{a}}{2\hslash}E\; \Delta_{a}} - {\left( {\frac{1}{T_{2}} - {\; \Delta_{\omega}}} \right){\eta_{a}.}}}} & (17)\end{matrix}$

Then the solutions for η_(a) and Δ_(a) change. Analytical solutions forη_(a) and Δ_(a) may be found in the presence of detuning Δ_(ω) when τ

T₂, so that the term proportional to 1/T₂ may be neglected in thepolarization equation. Then, the solutions for η_(a) and Δ_(a) become

$\begin{matrix}{{\eta_{a} = {{\frac{\Delta_{\omega}\tau}{1 + \left( {\Delta_{\omega}\tau} \right)^{2}}{sech}\; x} + {\frac{1}{1 + \left( {\Delta_{\omega}\tau} \right)^{2}}{sech}\; x\; \tanh \; x}}},} & \left( {18a} \right) \\{{\Delta_{a} = {{- 1} + {\frac{2}{1 + \left( {\Delta_{\omega}\tau} \right)^{2}}{sech}\; x}}},} & \left( {18b} \right)\end{matrix}$

where x=t/τ−z/ντ and Δ_(a0)=−1 at t→∞.

On physical grounds, it is apparent that the criterion for acceptabledetuning is that Δ_(ω)≦1/T₂, since τ≦T₂ and the bandwidth of the pulsein angular frequency is ˜τ⁻¹. If T₂ is 100 fs, and we demandconservatively that Δ_(ω)<1/T₂, then Δ_(ω)<10¹² s⁻¹, corresponding to anallowed frequency detuning of 1.6×10¹¹ Hz, which is 2% of the carrierfrequency of 8 μm light and is not overly demanding.

In order for the solution reported in Eq. (5) to be of any practicalinterest, it must be robust when μ_(a) differs from 2μ_(g), when T_(1g)and T_(1a) are on the order of a picosecond or less, when an initialpulse that is long compared to its final, stable duration is injectedinto the medium, and when the initial pulse area differs from the idealvalue of π in the gain medium and 2π in the absorbing medium. TheMaxwell-Bloch equations must be solved computationally to determine whathappens under these conditions. For computational analysis, we normalizeEq. (1). We define Ē=(μ_(g)T_(2g)/ h)E and we introduce the retardedtime t′=t−(n/c)z, the normalized time t=t/T_(2g), and the normalizeddistance z=lz, so that Eq. (1) becomes

$\begin{matrix}{{\frac{\partial\overset{\_}{E}}{\partial\overset{\_}{z}} = {{{- }\overset{\_}{g}\; \eta_{g}} - {i\frac{\overset{\_}{a}}{\left( {T_{2a}/T_{2g}} \right)\overset{\_}{\mu}}\eta_{a}} - {\frac{1}{2}\overset{\_}{E}}}},} & \left( {19a} \right) \\{{\frac{\partial\eta_{g}}{\partial\overset{\_}{t}} = {{\frac{}{2}\Delta_{g}\overset{\_}{E}} - \eta_{g}}},} & \left( {19b} \right) \\{{\frac{\partial\Delta_{g}}{\partial\overset{\_}{t}} = {{\left( {{\eta_{g}{\overset{\_}{E}}^{*}} - {\eta_{g}^{*}\overset{\_}{E}}} \right)} + \frac{\Delta_{g\; 0} - \Delta_{g}}{T_{1\; g}/T_{2g}}}},} & \left( {19c} \right) \\{{\frac{\partial\eta_{a}}{\partial\overset{\_}{t}} = {{\frac{}{2}\overset{\_}{\mu}\; \Delta_{a}\overset{\_}{E}} - \frac{\eta_{a}}{T_{2a}/T_{2g}}}},} & \left( {19d} \right) \\{{\frac{\partial\Delta_{a}}{\partial\overset{\_}{t}} = {{\; {\overset{\_}{\mu}\left( {{\eta_{a}{\overset{\_}{E}}^{*}} - {\eta_{a}^{*}\overset{\_}{E}}} \right)}} + \frac{\Delta_{a\; 0} - \Delta_{a}}{T_{1\; a}/T_{2\; g}}}},} & \left( {19e} \right)\end{matrix}$

where g=g/l, ā=a/l, and μ=μ_(a)/μ_(g).

In our simulations, we used different window sizes, depending on thepulse evolution, and we verified that the pulse intensities are alwayszero at the window boundaries. We spaced our node points 1-5 fs apart,and we chose a step size between 1 and 10 μm, depending on the materialparameters in the simulation. In each simulation these values wereconstant and we checked that reducing these values made no visibledifference in our plotted results. Finally, we verified by extending thepropagation length that we were following the pulses over a sufficientlylong length to reliably determine whether the pulses are stable or not.In FIG. 6, we show the limits of g and ā for stable operation withdifferent values of T₁/T₂ when T_(1g)=T_(1a)≡T₁ and T_(2g)=T_(2a)≡T. Webegin by assuming that a hyperbolic-secant-shaped pulse having an areaof π is injected into the QCL. Before the pulse is injected, thepopulation is completely in the upper lasing state in the gain medium,i.e., Δ_(g0)=1.0 and is completely in the ground state in the absorbingmedium, i.e., Δ_(a0)=−1.0. In FIG. 6, the black solid line at the bottomdefines the lower limits of ā for any T₁/T₂. The solid curves on the topare the loss-limited boundaries for different values of T₁/T₂. Theinjected pulses are only stable when the gain and absorption parametersare set between these two boundary limits. Stable pulses propagate inthe laser cavity with no change in shape and energy. We show the pulseevolution in the stable regime and unstable regimes in FIG. 7. FIG. 7 ashows stable pulse evolution when g=4.0 and ā=3.5. The laser becomesunstable when operated with ā smaller than the lower limits given inFIG. 6 due to the growth of continuous waves. In this case, the net gainof the laser becomes positive, i.e., g−ā−1>0, and multiple pulses mayform in the cavity, leading to the generation of multiple pulses in oursimulations. We give an example of this behavior in FIG. 7 b. In thiscase, we set g=4.0 and ā=1.0; the laser becomes unstable when z=20 andthe laser cavity develops more than one pulse. With a greater than theupper limit, the gain medium cannot compensate for absorption and thelinear loss. As a result, pulses damp. In FIG. 7 c, which exhibits thisbehavior, we have set g=4.0 and ā=7.8. The upper limit for ā decreaseswhen T₁/T₂ decreases as shown in FIG. 6, because the damping increases.We also show contours of the stable normalized pulse duration,τ=τ_(FWHM)/1.763T₂, with dashed lines in FIG. 6. Pulse durations are onthe order of T₂ when g≧2.5 and ā≧2.0. The pulse durations can be madearbitrarily short by increasing g and ā. However, g and ā areproportional to the current, so that the current must be increased. IfT₁/T₂ decreases, then g must increase if ā is constant in order tomaintain ti at a constant value.

As we discussed, in Sec. III, in a practical QCL design, we must haveT_(1g)>T_(1a). For generality, we consider here the stability limits asT_(1a)/T_(1g) varies between 0.5 and 2.0. FIG. 8 shows the stabilitylimits of g and ā as T_(1a)/T_(1g) is varied. The solid black line atthe bottom is the lower limit of ā and remains the same for anyT_(1a)/T_(1g). However, the upper limit of ā decreases whenT_(1a)/T_(1g) decreases. The analytical solution of the Maxwell-Blochequations given in Eq. (5) assumes that the absorbing medium has adipole moment twice that of the gain medium, i.e., μ_(a)=2μ_(g). Thecondition μ_(a)=2μ_(g) will not be exactly satisfied due to designconstraints and growth limitations. The QCL gain is determined by μ_(g)and T_(1g). To produce large gain, it is preferable that μ_(g) is large.In a vertical transition QCL, the dipole moment is generally 2 nm. Indiagonal-transition QCLs, the dipole moment is ≧1.4 nm, which issmaller. Despite the smaller value of μ_(g) with diagonal transitions,we must have μ_(a)/e≧2.8 to satisfy the condition μ_(a)=2μ_(g).Therefore, it is useful in practice if SIT mode locking is possible whenμ_(a)<2μ_(g). We determine the stability limits of μ=μ_(a)/μ_(g) forstable operation. FIG. 9 shows the lower and upper stability limits of μvs g as ā varies. The solid lines in FIG. 9 indicate the lower limitsfor μ while the dashed lines indicate the upper limits. The two ends ofeach of the lines are at the stability boundaries for g at eachparticular ā. In each of the cases, the minimum value of μ isapproximately 2 when g is near its minimum, below which an input pulseattenuates. As g increases toward the limit at which continuous wavesbecome unstable, the minimum value of μ required for stable operationdecreases significantly. Pulses are stable with μ˜1.2 when g=3.5, 4.0,and 4.5 with ā=2.5, 3.0, and 3.5, respectively, with g just below thestability limit for generating continuous waves. However, the stablepulse duration increases significantly as μ decreases. When μ is belowthe solid lines in FIG. 9, pulses attenuate. The minimum value of μrequired for stable operation increases as ā increases for any fixed g.We have found no hard upper limit to stability as μ increases, althoughthe pulses are increasingly distorted. The dashed lines in FIG. 9indicate the values of μ at which the pulses become double peaked. Wehave derived an energy balance equation in Eq. (16) that defines thelimits of the input energy for stable operation. However, Eq. (16)assumes that input pulse has a duration, τ_(i)=τ_(FWHM)/1.763

T₂

T₁, so that the effects of a finite coherence time T₂ and damping due tofinite T₁ may be ignored. If this condition is not satisfied, then Eq.(16) is no longer valid. From a practical standpoint, an input pulsehaving a duration on the order of T₂ or longer than T₂ is advantageous.We have calculated the dependence of the minimum and maximum inputenergy on the input pulse duration for two different combinations ofgain and loss. We show the results in FIG. 10. The input pulse durationis normalized to T_(2g)=T_(2a)≡T₂ and is plotted on a logarithmic scale.The value of T₁/T₂ has been set to 10. When τ_(i)/T₂=0.1, we find thatthe minimum normalized energy θ=(μ_(g)/ h)∫_(−∞) ^(∞)Edt=∫_(−∞) ^(∞)Ed tthat is required for stable operation is 0.30π when g=3.5 and ā=3.0.However, as we increase τ_(i)/T₂, the minimum normalized pulse energythat is required for stable operation increases significantly due to thepulse's decorrelation over its duration. It increases to 0.42π whenτ_(i)/T₂=1, 1.31π when τ_(i)/T₂=10, and 9.59π when τ_(i)/T₂=100. Pulsesare stable for an input energy of at least 20π when τ_(i)/T₂≦3. We findthat pulses split into multiple pulses when the input pulse energy is≧2π. However, at the stable pulse duration τ˜0.5 for the parametersg=3.5, ā=3.0, only one pulse is stable, and the others damp even with aninitial normalized energy of 20π when τ_(i)/T₂<4. When τ_(i)/T₂<4,continuous waves become unstable. We find that multiple pulses aregenerated when the input energy is ≧3π with τ_(i)/T₂=10. The upperstability limit for the input energy decreases as τ_(i)/T₂ increaseswhen τ_(i)/T₂≦10. However, beyond that point, the upper stability limitincreases with τ_(i)/T₂ as damping due to T₁ comes into effect. Thestability limits for the input normalized energy when g=3.5 and ā=3.5show a similar trend with the exception that both the stability limitsare shifted upward due to an increase in absorption. We simulated anumber of cases in which we investigated the effect of detuning theabsorbing medium from the gain medium and the carrier frequency of thelight. Setting T₁/T₂=10, Δ_(g0)=1.0, and Δ_(a0)=−1.0, we found thatstable operation can be obtained with a detuning Δ_(ω)T₂≦0.53 wheng=3.5, ā=3.5. Stable operation can be obtained with Δ_(ω)T₂≦0.36 wheng=3.5, ā=3.0, and with Δ_(ω)T₂≦0.15 when g=3.5, ā=2.5.

Accordingly, in the context of the above teaching, it has been shownthat by combining absorbing with gain periods in a QCL, one can createnearly ideal conditions to observe SIT mode locking and thereby obtainpulses that are less than 100 fs long from a mid-infrared laser. Here wealso show that this analysis can be extended using detailedcomputational studies of the Maxwell-Bloch equations, to illustrate thestability of the solutions as the equation parameters vary. Thesolutions demonstrate the robustness of the SIT mode-locking techniqueand that QCLs can be mode-locked using the SIT effect within practicallyachievable parameter regimes.

Any references recited or referred to herein are incorporated herein intheir entirety, particularly as they relate to teaching the level ofordinary skill in this art and for any disclosure necessary for thecommoner understanding of the subject matter of the claimed invention.It will be clear to a person of ordinary skill in the art that the aboveembodiments may be altered or that insubstantial changes may be madewithout departing from the scope of the invention. Accordingly, thescope of the invention is determined by the scope of the followingclaims and their equitable equivalents.

1. A self-induced transparency mode-locked quantum cascade laser,comprising: (i) an active section comprising a plurality of quantum welllayers deposited in alternating layers on a plurality of quantum barrierlayers and form a sequence of alternating gain and absorbing periods,said alternating gain and absorbing periods interleaved along the growthaxis of the active section, wherein the absorbing periods have a dipolemoment of about twice that of the gain periods; (ii) an optical cavitythat houses the active section and permits amplified light to escape;(iii) an externally supplied seed pulse; and (iv) current injectorsstructured and arranged to apply an electric control field to the activesection.
 2. The laser of claim 1, further comprising wherein the quantumbarrier layers are made using Indium-Aluminum-Arsenide, and the quantumwell layers are made using Indium-Gallium-Arsenide.
 3. The laser ofclaim 1, further comprising wherein the gain and absorbing periods aredesigned to provide a mid-IR wavelength laser.
 4. The laser of claim 1,further comprising wherein the gain and absorbing periods are designedto provide a mid-IR wavelength laser having a wavelength of betweenabout 8 micrometers and about 12 micrometers.
 5. The laser of claim 1,further comprising wherein the gain and absorbing periods are designedto provide coherence times

gain recovery times

round trip times.
 6. The laser of claim 1, further comprising whereinthe gain and absorbing periods are designed to provide coherence timesof at least 100 femtoseconds, gain recovery times of at least 1picosecond, and round trip times of at least 50 picoseconds.
 7. Thelaser of claim 1, further comprising wherein the gain and absorbingperiods are designed to provide a heterostructure having a ratio of 4gain periods:1 absorbing period.
 8. The laser of claim 1, furthercomprising wherein the input pulse is a it pulse in the gain medium. 9.The laser of claim 1, further comprising wherein each gain period andeach absorbing period have over 16 quantum layers.
 10. The laser ofclaim 1, further comprising wherein the gain and absorbing periods aredesigned to suppress continuous waves and eliminate spatial holeburning.
 11. The laser of claim 1, further comprising wherein the gainand absorbing periods are designed to provide a mid-IR wavelength laserhaving pulse length less than 100 femtoseconds long.